**1. Introduction**

Carboxymethyl cellulose (CMC), also known as cellulose gum [1], has many features: a high solubility, clarity of its solutions, the ability to hold water, controlled crystal growth, and it can modify viscosity, in addition to its capacity to fit the required smooth texture or body. These multifunctional aspects of a non-toxic cellulose derivative are why it is utilized in many industries and technical applications. It is employed to enhance moisturizing impact due to its polymeric structure that works as a film-forming factor [2,3]. CMC is utilized in paper industries and pharmaceuticals and is also used to stabilize clay particles [2,4] and others [5–10]. In view of the massive uses of CMC, many researchers have devoted their time to studying it. Saqib et al. [11,12] employed a Caputo–Fabrizio fractional derivative (CFFD) approach and an Atangana–Baleanu fractional derivative (ABFD) approach alongside the Laplace technique to investigate the convection flow of CMC-water nanofluid. They confirmed that multiple wall carbon nanotubes are more effective in terms of improved heat transfer, and that the velocity of CMC-water is higher with multiple wall carbon nanotubes. Rahmati et al. [13] examined the laminar flow of a CMC-aqueous solution in a horizontal 2D microtube. Their findings revealed that

the slip velocity coe fficient contributed notably to the growth of the heat transfer rate, and significantly reduced the friction factor of the horizontal microtube wall.

The real reason for using nanotechnology is its capacity to work at the molecular level, atom-by-atom, to make large structures via essentially novel molecular organization. The actual birth of nanotechnology was at the end of 1959 when it was introduced by physicist Richard P Feynman [14]. He concluded that the physical properties of materials change depending on the scale of its molecules, and also posed two challenges: writing "Encyclopedia Britannica" on the head of a pin and making the nanometer. Two decades later, IBM Zurich scientists were able to invent the scanning tunneling microscope, which enabled scientists for the first time to observe materials at the atomic scale, a paradigm shift that had significantly contributed to the spread of nanotechnology in all industrialized countries by the 1990s. In the heat transfer field, Choi and Eastman [15] incorporated nanotechnology unprecedentedly through immersed metallic nanoparticles in a base fluid. These ultrafine particles possessed extraordinary properties that made them notably improve the thermal conductivity of the ordinary fluid. Buongiorno [16] developed a mathematical model that shows that the heat transfer rate is a ffected by several factors other than the thermal conductivity impact. Tiwari and Das [17] also developed a mathematical model to consider the solid volume fraction. Recently, many researchers have used the Tiwari and Das model to examine the nanofluid flow behavior of nanoparticles. Swalmeh et al. [18] used the Tiwari and Das model to investigate the behavior of micropolar nanofluid from a sphere. Selimefendigil et al. [19] analyzed the magnetohydrodynamic (MHD) combined convection flow of a nanofluid in a lid-driven triangular cavity by the use of the Tiwari and Das model. Alwawi et al. [20] employed the Tiwari and Das model to simulate the flow behavior of a sodium alginate based Casson nanofluid from a sphere. Metal nanoparticles are distinguished by excellent electrical and thermal conductivity, chemical stability, optical and magnetic distinct properties and also, they have a high surface-to-volume ratio. However, in this study aluminum (Al), copper (Cu), and silver (Ag) metal nanoparticles were used because of their similar thermo-physical properties and their common uses and many applications in polymers and pharmaceuticals [21–23], which may be due to their presence accompanied with the presence of CMC-water in these applications.

In real life, mixed convection plays a pivotal role in many engineering and industrial applications. It appears clearly in the cooling of electronic devices and nuclear reactors, food processing, and solar collectors. In addition, Lorentz forces, generated by the passage of a magnetic field via a flowing conducting fluid, has occupied a prominent place in several modern processes of metallurgy and metalworking. Makinde and Aziz [24] analyzed mixed convection on a vertical plate in a porous medium considering the MHD impact and convective boundary condition. Tham et al. [25] studied the boundary layer flow of nanofluid with the MHD e ffect. Chamkha et al. [26] investigated the magneto-mixed convection flow of ferrofluids in the presence of a partial slip. Here are some of the most important recently conducted studies related to MHD mixed convection [27–32].

Casson's model [33] was developed in 1959 to be able to predict the behavior of non-Newtonian fluids e fficiently, and since then it has demonstrated its competence by foretelling the behavior of shear-thinning fluids, such as human blood, honey, concentrated fruit juice, ketchup, and others. Later a considerable number of articles employed this model. Malik et al. [34] employed the Runge–Kutta–Fehlberg technique to examine the flow of a Casson nanoliquid about a vertical cylinder. Mukhopadhyay et al. [35] emphasized that the flow separation could be curbed by raising the Casson parameter. Mustafa et al. [36] investigated the convection of Casson fluid from a stretching sheet taking into account viscous dissipation. See also these recent and e fficient studies [37–41].

To the best of our knowledge, and judging by the prior literature, no study has been conducted on the heat transfer of a CMC-based Casson nanoliquid induced by combined convection past a solid sphere with a MHD influence via the KBM that has been investigated in this work. It is also an extension and development of these studies [20,25,42–44] which may be useful in academic studies, polymer processes, pharmaceutical and food industries, and others.
